Related papers: Mean oscillation conditions for nonlinear equation…
We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) -…
We establish a Liouville type theorem for fully nonlinear uniformly elliptic equations in exterior domains in half spaces under quadratic boundary data and a quadratic growth condition, that is, any viscosity solution tends to a quadratic…
We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic $\mathcal{N}$-function.…
In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$ \left\{ \begin{array}{ll} -\Delta u+u=|u|^{r-2}u &\text{in} \; \Omega,\\ \\ \frac{\partial…
In this paper, we study the following quasi-linear elliptic inequality $\Delta_m u +u^p |\nabla u|^q \leqslant 0$ on weighted graphs, where $(m,p,q)\in (1,\infty)\times\mathbb{R}\times\mathbb{R}$. According to the ranges of parameters $(m,…
In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: $ \left\{ \begin{array}{ll} -\mathcal Q_{p}u =[H(Du)]^{q}+f(x)…
We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \begin{equation*} -{\rm div} A(x,\nabla u)= f\in L^1(\Omega), \end{equation*} on…
Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega \times(0,T).$ We study problems of the model type \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u=\mu\qquad\text{in }Q,\\ {u}=0\qquad\text{on…
We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is…
Consider \[ \begin{cases} F(D^2 u,Du,u,x) = u^{-p}v^{-q},~\text{in}~\Omega\\ F(D^2 v,Dv,v,x)=u^{-r}v^{-s},~~\text{in}~~\Omega\\ u,v>0~~\text{in}~~\Omega\\ u=v=0~\quad~\text{on}~~\partial\Omega, \end{cases} \] where $\Omega$ is an open…
We consider the homogeneous Dirichlet problem for the anisotropic parabolic equation \[ u_t-\sum_{i=1}^ND_{x_i}\left(|D_{x_i}u|^{p_i(x,t)-2}D_{x_i}u\right)=f(x,t) \] in the cylinder $\Omega\times (0,T)$, where $\Omega\subset \mathbb{R}^N$,…
This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{…
We study the existence problem for positive solutions $u \in L^{r}(\mathbb{R}^{n})$, $0<r<\infty$, to the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n \] in the sub-natural growth case…
This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form $$-div (|x|^{a} D u ) = f(x,u), ~ u > 0,\, \mbox{ in } \Omega,$$ where $N \geq 3$, $\Omega$ is an open domain in…
In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u,x)=f$ in $\mathbb{R}^n$, where $f$ is periodic and $F$ has the periodicity in $x$. Under the assumption that the oscillation of…
We prove existence and regularity results for weak solutions of non linear elliptic systems with non variational structure satisfying $(p,q)$-growth conditions. In particular we are able to prove higher differentiability results under a…
Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic…
We investigate solutions to nonlinear elliptic Dirichlet problems of the type \[ \left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= \mu &\qquad \mathrm{ in}\qquad \Omega, u=0 &\qquad \mathrm{ on}\qquad \partial\Omega, \end{array}\right.…
In this paper, we obtain $C^{1,\alpha}$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2} \nabla u +…
In the present paper, we study sharp C^{1;\alpha} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.